Let \(g\ge 1\)  be an integer and let \((C, x_1, \cdots , x_n)\)  be an analytically
general hyperbolic \(n\) -pointed curve of genus \(g\) . Let \(U\subset C\setminus \lbrace x_1, \cdots , x_n\rbrace \)  be a dense Zariski-open subset. Let \(f: Y\rightarrow U\)  be a smooth proper morphism, \(i\ge 0\)  an integer, and suppose \(\mathbb {V}\)  is a complex local system on \((C, x_1, \cdots , x_n)\)  with infinite monodromy such that \(\mathbb {V}|_U\)  is a summand of \(R^if_*\mathbb {C}\) . Then we wish to show that \(\dim _{\mathbb {C}}\mathbb {V}\ge 2\sqrt{g+1}\) .
It suffices to show that \(\mathbb {V}\)  satisfies the hypotheses of
theorem:very-general-VHS. The existence of an
\({O}_K\) -structure follows from the fact that \(R^if_*\mathbb {C}\)  has a
\(\mathbb {Z}\) -structure.
Let \(\mathbb {W}\)  be the corresponding \({O}_K\) -local system. All that remains is to verify that for each embedding
\(\iota :{O}_K\hookrightarrow \mathbb {C}\) , the corresponding complex local
system \(\mathbb {W}_\iota :=\mathbb {W}\otimes _{{O}_K, \iota } \mathbb {C}\) 
underlies a polarizable complex variation of Hodge structure. But each such
embedding yields a summand \(\mathbb {W}_\iota |_U\)  of \(R^if_*\mathbb {C}\) ,
Galois-conjugate to the original embedding \(\mathbb {W}|_U\subset \mathbb {V}|_U\subset R^if_*\mathbb {C}.\)  Any such summand underlies a polarizable
complex variation of Hodge structure, by prop:basic-facts(2). Now
\(\mathbb {W}_\iota |_U\)  extends from \(U\)  to \(C\setminus \lbrace x_1, \cdots , x_n\rbrace \)  and
so the same is true for the corresponding complex polarizable variation of Hodge
structure by [1]}, which completes the proof.
